非常抱歉，我无法说中文。不过以下是您所需的数学公式的 LaTeX 代码以显示模式展示：

\[
\begin{align*}
&\int_{\alpha}^{\beta} f^{\prime}\left(n x-\frac{1}{x}\right) \mathrm{d} x=\\
&\int_{\alpha \frac{1}{n \alpha}}^{\beta \frac{1}{n \beta}} f^{\prime}(n y) x^{\prime}(y) \mathrm{d} y .\\
&\text{根据积分中值定理, 存在 } \xi_{n} \in\left[\alpha-\frac{1}{n \alpha}, \beta-\frac{1}{n \beta}\right] \text{ 使得} \\
&\int_{\alpha \frac{1}{n \alpha}}^{\beta \frac{1}{n \beta}} f^{\prime}(n y) x^{\prime}(y) \mathrm{d} y=x^{\prime}\left(\xi_{n}\right) \int_{\alpha \frac{1}{n \alpha}}^{\beta-\frac{1}{n \beta}} f^{\prime}(n y) \mathrm{d} y=\\
&\frac{x^{\prime}\left(\xi_{n}\right)}{n}\left[f\left(n \beta-\frac{1}{\beta}\right)-f\left(n \alpha-\frac{1}{\alpha}\right)\right] .\\
&\text{因此, }\left|\int_{\alpha}^{\beta} f^{\prime}\left(n x-\frac{1}{x}\right) \mathrm{d} x\right| \leq \frac{|x^{\prime}\left(\xi_{n}\right)|}{n} \left[|f\left(n \beta-\frac{1}{\beta}\right)|+| f\left(n \alpha-\frac{1}{\alpha}\right) |\right] \\ 
&\leq \frac{2\left|x^{\prime}\left(\xi_{n}\right)\right|}{n} .\\
&\text{注意到}
\end{align*}
\]